Semi-analytical models (SAMs) describe the physical processes regulating galaxies' evolution (e.g., star formation, BH growth, chemical enrichment of the ISM, stellar/BH feedback) by means of a set of analytic equations (called prescriptions), linked to the hierarchical assembly history of structure formation.
In this approach, the hierarchical assembly, trough cosmic times, of massive halos hosting QSOs (the so-called merger trees, or merger history) is reconstructed either analytically (e.g., using Monte Carlo algorithms) on the basis of formalism describing the halo mass function at different times (e.g., [1]} and references therein) or by means
of N\(-\) body, dark matter only, cosmological simulations of large volumes of the Universe (\(>100 \, \rm Mpc^{3}\)  comoving; e.g., [2]}).
